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                        <h1 style="font-size: 48px;">数学复习之高等数学（一）：极限与连续</h1>
                        <p class="post-info" style="margin-bottom: 15px;">
                            发表于:
                            <span style="color: #aaa; margin: 0 15px">
                                2018.08.01
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                             | 分类：
                             <a href="#" style="margin: 0 5px">
                                
                                    数学
                                
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                                    <a href="#" class="post-tag">数学</a>
                                
                            
                        </p>
                        <h3 id="主要概念"><a href="#主要概念" class="headerlink" title="主要概念"></a>主要概念</h3><ol>
<li><p>数列的极限 （$\epsilon-N$）</p>
<p>如果 $\forall\epsilon&gt;0 ,\exists N&gt;0$，对任意的 $n&gt;N$， 存在 $\vert a_n - A \vert &lt; \epsilon$，则称数列 $a_n$收敛于A。</p>
<p>记为 $$\lim_{n \to \infty} a_n = A$$</p>
</li>
<li><p>函数的极限 （$\epsilon - \delta$）</p>
<p>如果 $\forall \epsilon &gt; 0, \exists \delta &gt; 0$，对任意的 $ 0 &lt; \vert x - a \vert &lt; \delta$，都有 $\vert f(x) - A \vert &lt; \epsilon$，那么称A为函数在趋近于a时的极限。记为 $$ \lim_{n \to a} f(x) $$</p>
</li>
<li><p>X的极限（$\epsilon - X$）</p>
<p>如果 $\forall \epsilon &gt; 0, \exists X &gt; 0$ ,对任意的有 x &gt; X, 存在 $\vert f(x) - A \vert &lt; \epsilon$， 那么称 $f(x)$趋于A。</p>
<p>记为 $$\lim_{x \to \infty} f(x) = A$$</p>
</li>
<li><p>无穷小</p>
<p>以0为极限就叫做无穷小，两个无穷小之间不能相减，可以相除。</p>
</li>
<li><p>极限的本质：<strong>无限的接近而不相等</strong></p>
<p>如果 $\forall \epsilon &gt; 0, \exists \delta &gt; 0$，对任意的 $ 0 &lt; \vert x - a \vert &lt; \delta$，都有 $\vert f(x) - A \vert \le 3\epsilon$ ， 同样为函数极限的定义。</p>
</li>
<li><p>连续：极限值等于函数值就叫连续。</p>
<blockquote>
<p>$$f(x)在x=a连续 \Leftrightarrow f(a-0) = f(a+0) = f(a)$$</p>
</blockquote>
</li>
<li><p>闭区间连续</p>
<figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br></pre></td><td class="code"><pre><span class="line">如果f(x)在(a,b)内处处连续，f(a)=f(a+0)，f(b)=f(b-0)</span><br><span class="line">则称：f(x)在[a,b]上连续，记为f(x)属于c[a,b]</span><br></pre></td></tr></table></figure>
</li>
<li><p>间断</p>
<p>$$如果lim_{x \to a} f(x) \neq f(a)，称x=a为f(x)的间断点$$</p>
<blockquote>
<p>第一类间断点：$f(a-0),f(a+0) \exists$</p>
</blockquote>
<p>$$\begin{cases} f(a-0) = f(a+0) \neq f(a)&amp; a为可去间断点 \\ f(a-0) \neq f(a+0)&amp;a为跳跃间断点 \end{cases}$$</p>
<blockquote>
<p>第二类间断点：$f(a-0),f(a+0)$至少有一个不存在</p>
</blockquote>
</li>
</ol>
<a id="more"></a>
<h3 id="重点知识"><a href="#重点知识" class="headerlink" title="重点知识"></a>重点知识</h3><ol>
<li><p>极限的性质</p>
<ul>
<li><p>唯一性</p>
</li>
<li><p>保号性</p>
<p>口诀：<strong>极限正，则去心邻域正，极限负，则去心邻域负。</strong></p>
</li>
<li><p>有界性：单调递增，有下界，无上界则无极限，有上界则有极限。单调递减，有上界，有下界则有极限，无下界则无极限。</p>
<p>证明单调函数极限存在时的口诀：<strong>单调递增找上界，单调递减找下界。</strong></p>
</li>
</ul>
</li>
</ol>
<p>例1:</p>
<p>$$f’(x)=0, \lim_{x \to 0}  \frac {f’(x)}{x^3}=2， 请问x=0为原函数的什么点？$$</p>
<p>解：</p>
<p>$$\because \lim_{x \to 0}  \frac {f’(x)}{x^3}=2 &gt; 0$$</p>
<p>$$\therefore \exists \delta &gt; 0, 使得 0 &lt; \vert x - 0 \vert &lt; \delta $$</p>
<p>$$\begin{cases} f’(x) &lt; 0, x \in (-\delta, 0) \\ f’(x) &gt; 0, x \in (0, \delta) \end{cases} $$</p>
<p>$$\therefore 所以 x = 0 为极小点。$$</p>
<ol>
<li><p>极限的存在性证明</p>
<ul>
<li><p>准则1：夹逼定理</p>
<p>描述：</p>
<p>$$如果a_n \le b_n \le c_n， 同时\lim_{n \to \infty} a_n = \lim_{n \to \infty} c_n = A 则 \Rightarrow \lim_{n \to \infty} b_n = A$$</p>
<blockquote>
<p>证明：</p>
<p>$$\because \lim_{n \to \infty} a_n = \lim_{n \to \infty}c_n A$$</p>
<p>$$即 A-\epsilon &lt; a_n&lt; A+\epsilon$$</p>
<p>$$\because a_n \le b_n$$</p>
<p>$$\therefore A-\epsilon &lt; a_n \le b_n$$</p>
<p>$$同理，b_n \le c_n &lt; A+\epsilon$$</p>
<p>$$即A-\epsilon &lt; b_n &lt; A+\epsilon$$</p>
<p>$$即证\lim_{n \to \infty} b_n = A$$</p>
</blockquote>
<p>使用场景：当分子或分母次数不齐的时候使用。</p>
</li>
<li><p>准则2：定积分</p>
<p>描述：</p>
<p>$$\lim_{n \to \infty} \frac 1n \sum_{i=1}^n f(\frac in) = \int_0^1 f(x) dx $$</p>
<p>使用场景：分子分母次数齐的时候使用。</p>
</li>
</ul>
</li>
<li><p>无穷小的性质</p>
<ol>
<li>一般性质<ul>
<li>$$\alpha \to 0, \beta \to 0 \Rightarrow \begin{cases} \alpha + \beta \to 0 \\ \alpha \cdot \beta \to 0 \\ k \alpha \to 0 \end{cases}$$</li>
<li>$$\vert \alpha \vert \le M, \beta \to 0 \Rightarrow \alpha \cdot \beta \to 0$$</li>
<li>$$\lim_{x \to \infty} f(x) = A \Leftrightarrow f(x) = A + \alpha$$</li>
</ul>
</li>
<li>等价性质<ul>
<li>$$\begin{cases} \alpha \sim \alpha \\ \alpha \sim \beta \Rightarrow \beta \sim \alpha \\ \alpha \sim \beta, \beta \sim \gamma \Rightarrow \alpha \sim \gamma \end{cases}$$</li>
<li><strong>重要：</strong>$\alpha \sim \alpha’, \beta \sim \beta’ 且 \lim \frac {\beta’}{\alpha’} = A \Rightarrow \lim \frac \beta\alpha = A$</li>
</ul>
</li>
<li>$x \to 0:$<ul>
<li>$$x \sim sinx \sim tanx \sim arcsinx \sim arctanx \sim e^x-1 \sim ln(1+x)$$</li>
<li>$$1-cosx \sim \frac 12 \cdot x^2$$</li>
<li>$$(1+x)^a -1 \sim a \cdot x$$</li>
</ul>
</li>
</ol>
</li>
</ol>
<h3 id="两个重要极限"><a href="#两个重要极限" class="headerlink" title="两个重要极限"></a>两个重要极限</h3><ol>
<li>$$\lim_{\Delta \to 0} \frac {sin\Delta}{\Delta} = 1$$</li>
<li>$$\lim_{\Delta \to 0} (1 + \Delta)^{\frac 1\Delta} = e$$</li>
</ol>
<h3 id="不定型"><a href="#不定型" class="headerlink" title="不定型"></a>不定型</h3><ol>
<li>$\frac 00 型$<ul>
<li>$$u(x)^{v(x)} \Rightarrow e^{v(x) \cdot lnu(x)}$$</li>
<li>$$(\quad) - 1 \Rightarrow \begin{cases} e^\Delta-1 \sim \Delta \\ (1+\Delta)^a-1 \sim a\cdot\Delta (\Delta \to 0) \end{cases}$$</li>
<li>$$ln(\quad) \Rightarrow ln(1+\Delta) \sim \Delta (\Delta \to 0)$$</li>
</ul>
</li>
<li>$1^\infty$<ul>
<li>$$凑 (1 + \Delta)^{\frac 1\Delta}$$</li>
<li>$$恒等变形$$</li>
</ul>
</li>
<li>$\frac \infty\infty$<ul>
<li>洛必达法则</li>
<li>转化为确定型</li>
<li>$$\lim_{x \to \infty} \frac{b_nx^n+\cdots +b_0}{a_nx^m+ \cdots + a_0} \Rightarrow \begin{cases}<br> =0, n&lt;m\\<br> \infty, n&gt;m\\<br> \frac {b_n}{a_n}, n=m<br> \end{cases}​$$</li>
</ul>
</li>
<li>$\infty - \infty$<ul>
<li>有分母则通分，无分母则转化（共轭函数）</li>
</ul>
</li>
</ol>
<h3 id="闭区间连续的性质"><a href="#闭区间连续的性质" class="headerlink" title="闭区间连续的性质"></a>闭区间连续的性质</h3><blockquote>
<p>$f(x) \in c[a,b]$</p>
</blockquote>
<ul>
<li>有最大值M和最小值m</li>
<li>有上界和下界</li>
<li>零点定理：$f(a) \cdot f(b) &lt; 0 \Rightarrow \exists c \in (a,b)$,使$f(c)=0$</li>
<li>介值定理：$f(x) \in c[a,b], \forall \eta \in [m,M],\exists \zeta \in [a,b], 使f(\zeta) = \eta$ （即是说，位于m和M之间的值f(x)皆可取到。）</li>
</ul>
<blockquote>
<p>何时使用零点定理和介值定理</p>
</blockquote>
<ol>
<li>$$f(x) \in c[a,b], \eta \in 开区间 \Rightarrow 零点定理$$</li>
<li>$$f(x) \in c[a,b],若 \begin{cases} \eta[a,b] \\ 函数值之和 \end{cases} \Rightarrow 介值定理$$</li>
</ol>

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